I can't resist referring to the paper

R. Brown and P.J. Higgins, ``On the connection between the second relative homotopy groups of some related spaces'', {\em Proc. London Math. Soc.} (3) 36 (1978) 193-212,

which gives some explicit calculations with pushouts of crossed modules, but which is little referred to, and further examples are in

(with C.D.WENSLEY), `Computation and homotopical applications of induced crossed modules', J. Symbolic Computation 35 (2003) 59-72.

The key to the proof of the theorem is the notion of homotopy double groupoid of a based pair.

The paper arXiv:0909.3387v2 gives applications of higher van Kampen theorems to homotopy groups of spheres.

January 22, 2016 Tom refers to "an extra bit of structure". This structure is crucial for applications of a colimit theorem (see the above 1978 paper, and the 2011 book partially titled 2011 book Nonabelian Algebraic Topology, NAT, pdf available); altering the structure alters the colimits. Part I of this book gives lots of explicit calculations, for example showing how for $A$ connected, and $f:A \to X$ a pointed map, the crossed module $$\pi_2(X \cup _f CA,X,x) \to \pi_1(X,x)$$is determined by the morphism $f_*:\pi_1(A) \to \pi_1(X)$. This is a generalisation of a theorem of JHC Whitehead in "Combinatorial Homotopy II" on free crossed modules, which is the case $A$ is a wedge of circles.

There is a general assumption that we most want to calculate the second homotopy group $\pi_2$; but this, even as a module over $\pi_1$ is but a pale shadow of the 2-type.

One of the problems in homotopy theory is that identifications in low dimensions have influence on high dimensional invariants. So part of the aim is to study this influence using algebraic structures which have information in a range of dimensions, and to develop and apply nonabelian colimit theorems in higher homotopy.

Tim Porter refers to the nonabelian tensor square, which generalises to tensor products. I have kept up a bibliography on this topic; the idea arose from considering pushouts of crossed squares, and the bibliography currently has 144 items, dating back to 1952.

January 23,2016 I thought of another point which maybe answers the question better, namely as to the import of these crossed module rules on this boundary map for second relative homotopy groups. The situation becomes clearer if you work with the homotopy double groupoid of a pointed pair of spaces, see the NAT book and also the presentation at Galway on my preprint page. In terms of double groupoids, the first rule is a boundary rule for a certain subdivided square, and the second rule is equivalent to the interchange law for the two compositions. So if you draw the right pictures, then these rules become necessities.

The emphasis on homotopy **groups** is part of the "squashing" of 2-dimensional situation into a single dimension, on a line. Then some natural rules become obscure, and it becomes impossible to do the 2-dimensional compositions which are necessary for the proof of the Seifert-van Kampen Theorem for homotopy double groupoids and so for the homotopy crossed module of the question.

Topology and Groupoids(2006). It's available in e-book form for 5 pound sterling at store.kagi.com/cgi-bin/store.cgi?storeID=6FEPD_LIVE . – Harry Gindi Aug 18 '10 at 17:22